Ever arrived early at a new destination in the morning after a long red-eye flight and not had a hotel room where you could crash?

It happened to me on my last trip, when I landed in Reykjavik, Iceland, at 6:30 a.m. on the same day I was scheduled to board a cruise ship bound for Greenland. The ship’s reception staff were willing to hold my suitcases for me, but my cabin wouldn’t be ready to enter until late afternoon. That left me with most of the day to explore the city — if I could stay awake that long.

The morning started well, with a stroll along the waterfront and visits to the city’s most famous church, Hallgrimskirkja, and its avant-garde new concert hall, Harpa. But the combination of jet lag and a long, sleepless flight the night before had me drooping after a few hours.

As long as I was walking outside in the fresh air and sunlight, I could stay relatively alert. Any time I sat down, though — at a restaurant for lunch, on a pew in Hallgrimskirkja to listen to the organist practice, or, worst of all, inside a darkened theater to watch a timelapse photography presentation on the northern lights — my eyelids got heavy and my chin drifted inexorably toward my chest until I jerked myself awake again. As the hours wore on, my pleasurable day of sightseeing turned into a forced march through the city streets until I could board my ship and finally, finally take a nap.

Even if you have a hotel room booked for the night, it’s not uncommon to arrive before the official check-in time and be told there simply aren’t any rooms available yet. One alternative is to try to find a day room where you can nap and shower. I recently discovered Between9and5.com, a site dedicated to hotels with same-day check-in and check-out. Unfortunately, offerings in some cities can be slim. There’s currently only one option in Reykjavik, and it starts at more than $200 a night — more than I was willing to pay for just a few hours.

Short of sleeping on a park bench, what do you do to stay awake and make the most of your first day when you’re exhausted after a flight?